C_4 decomposition of the tensor product of complete graphs - Opeyemi Oyewumi; Abolape Deborah Akwu (Electronic Journal of Graph Theory and Applications (EJGTA), 2020)
Let G be a simple and finite graph. A graph is said to be decomposed into subgraphs H1 and H2 which is denoted by G = H1 ⊕ H2, if G is the edge disjoint union of H1 and H2. If G = H1 ⊕ H2 ⊕ H3 ⊕ ... ⊕ Hk, where H1, H2, H3, ..., Hk are all isomorphic to H, then G is said to be H-decomposable. Futhermore, if H is a cycle of length m then we say that G is Cm-decomposable and this can be written as Cm|G. Where G × H denotes the tensor product of graphs G and H, in this paper, we prove the necessary and sufficient conditions for the existence of C4-decomposition of Km × Kn. Using these conditions it can be shown that every even regular complete multipartite graph G is  C4-decomposable if the number of edges of G is divisible by 4.